Theorem bnj91 31448

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

Hypotheses

Ref	Expression
bnj91.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj91.2	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
bnj91	⊢ ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑓   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝑍(𝑥,𝑦,𝑓)

Proof of Theorem bnj91

Step	Hyp	Ref	Expression
1		bnj91.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2	1	sbcbii 3689	. 2 ⊢ ([𝑍 / 𝑦]𝜑 ↔ [𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3		bnj91.2	. . 3 ⊢ 𝑍 ∈ V
4	3	bnj525 31325	. 2 ⊢ ([𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5	2, 4	bitri 267	1 ⊢ ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 198   = wceq 1653   ∈ wcel 2157  Vcvv 3385  [wsbc 3633  ∅c0 4115  ‘cfv 6101   predc-bnj14 31274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-sbc 3634

This theorem is referenced by:  bnj118  31456  bnj125  31459