Theorem bnj90 35120

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj90.1	⊢ 𝑌 ∈ V

Assertion

Ref	Expression
bnj90	⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌)

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝑌(𝑥, 𝑧)

Proof of Theorem bnj90

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj90.1	. 2 ⊢ 𝑌 ∈ V
2		fneq2 6627	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑦))
3		fneq2 6627	. . 3 ⊢ (𝑦 = 𝑌 → (𝑧 Fn 𝑦 ↔ 𝑧 Fn 𝑌))
4	2, 3	sbcie2g 3783	. 2 ⊢ (𝑌 ∈ V → ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌))
5	1, 4	ax-mp 5	1 ⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2142  Vcvv 3454  [wsbc 3743   Fn wfn 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3744  df-fn 6539

This theorem is used by:  bnj121  35267  bnj130  35271  bnj207  35278