Theorem bnj62 35050

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj62	⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem bnj62

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3467	. . . 4 ⊢ 𝑦 ∈ V
2		fneq1 6624	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴))
3	1, 2	sbcie 3794	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴)
4	3	sbcbii 3809	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑦]𝑦 Fn 𝐴)
5		sbccow 3776	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑥]𝑥 Fn 𝐴)
6		vex 3467	. . 3 ⊢ 𝑧 ∈ V
7		fneq1 6624	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴))
8	6, 7	sbcie 3794	. 2 ⊢ ([𝑧 / 𝑦]𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴)
9	4, 5, 8	3bitr3i 304	1 ⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴)

Syntax hints:   ↔ wb 209  [wsbc 3753   Fn wfn 6528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-fun 6535  df-fn 6536

This theorem is referenced by:  bnj156  35058  bnj976  35107  bnj581  35237