Theorem bnj62 34863

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 3433	. . . 4 ⊢ 𝑦 ∈ V
2		fneq1 6589	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴))
3	1, 2	sbcie 3770	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴)
4	3	sbcbii 3785	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑦]𝑦 Fn 𝐴)
5		sbccow 3751	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑥]𝑥 Fn 𝐴)
6		vex 3433	. . 3 ⊢ 𝑧 ∈ V
7		fneq1 6589	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴))
8	6, 7	sbcie 3770	. 2 ⊢ ([𝑧 / 𝑦]𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴)
9	4, 5, 8	3bitr3i 301	1 ⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴)

Syntax hints:   ↔ wb 206  [wsbc 3728   Fn wfn 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6500  df-fn 6501

This theorem is referenced by:  bnj156  34871  bnj976  34920  bnj581  35050